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Historical, Generic and Current Challenges of Adaptive Control

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Title: Historical, Generic and Current Challenges of Adaptive Control


1
Historical, Generic and Current Challenges of
Adaptive Control

2
Outline
  • Introduction
  • Some old problems of adaptive control
  • Generic and conceptual challenges
  • Topical Problems
  • Conclusions

3
Macro-view of presentation
  • Show some big mistakes in Adaptive Control
    history, especially by theoreticians
  • Show some pervasive adaptive control problems
    still not properly addressed
  • Show some newer developments considered-- with
    continuing regrettable inadequacies

4
Thanks
  • Recent/current collaborators
  • Alexander Lanzon
  • Andrea Lecchini
  • Michel Gevers
  • Xavier Bombois
  • Franky De Bruyne
  • Steve Morse
  • Thomas Brinsmead
  • Sunghan Cha
  • Michael Rotkowitz
  • Past collaborators
  • Rick Johnson
  • Iven Mareels
  • Bob Bitmead
  • Robert Kosut
  • Petar Kokotovic
  • Richard Johnstone
  • Shankar Sastry
  • Marc Bodson
  • Wee Sit Lee

and to the ALCOSP 2007 organizers for inviting
me!
5
Outline
  • Introduction
  • Some old problems of adaptive control
  • What is adaptive control?
  • Bursting
  • Rohrs Counterexample
  • MIT Rule
  • Iterative identification and control
  • Generic and conceptual challenges
  • Topical Problems
  • Conclusions

6
Adaptive Control
  • Plant is initially unknown or partially known, or
    is slowly varying.
  • There is an underlying performance
    index/optimization, e.g.

7
Adaptive Control (continued)
  • A non-adaptive controller maps the error signal
    r-y into u in a causal, time-invariant way e.g.
  • An adaptive controller is one where parameters
    are adjusted.

8
Possible Form of Adaptive Controller
  • Often 3 time scales
  • Underlying plant dynamics (with fixed parameters)
  • Time scale for identifying plant
  • Time scale of plant parameter variation
  • Nonlinearity is present!

9
Possible Form of Adaptive Controller
And now to the problems Bursting Rohrs
Counterexample Iterative Identification and
controller redesign MIT rule
  • Often 3 time scales
  • Underlying plant dynamics (with fixed parameters)
  • Time scale for identifying plant
  • Time scale of plant parameter variation
  • Nonlinearity is present!

10
Bursting Phenomenon
  1. Plant output should follow unit step input.
  2. Plant output yk well behaved till time 3400
  3. Recovery occurs by time 3500

11
Bursting Phenomenon
  • Bursting phenomena can come after 1 week. Their
    existence had not been predicted
  • Why do they occur? How could they be stopped?
  • From measurements of u(), y(), one should be
    able to identify b and c
  • If u constant, can only identify b/c-the DC
    gain
  • Adaptive controllers contain identifier of b
    and c

12
Bursting Phenomenon (Explanation)
  • Control law is designed based on estimates of b
    and c ? can accidentally implement unstable
    closed loop.
  • Instability then enriches the signals, giving
    improved identification. Stabilising control law
    is reapplied.
  • Practical issue must either turn off adaptation
    or drive the system with rich input.
  • Set-point control does not provide a rich input!

13
Rohrs Counterexample
  • Ingredients
  • A true system with high frequency dynamics
    uncaptured by a low complexity model
  • Nonpersistently excited input
  • An adaptive control task
  • Observation Instability
  • Claim of authors
  • All adaptive control theorems require an SPR
    condition
  • Unmodelled high frequency dynamics must destroy
    SPR
  • Therefore, no adaptive control system can ever
    work!!!
  • Explanation (not offered by the authors)
  • Mixture of MIT-rule like error (later) and
    bursting

14
Iterative Identification Controller Redesign
  • A frequently advanced approach to adaptive
    control design is iterative identification and
    controller redesign.
  • One iteration comprises
  • (re) identifying the plant with the current
    controller
  • redesigning the controller to achieve the design
    objective on the basis of the identified model,
  • implementing it on the real plant
  • This is like adaptive control, with a long wait
    between controller updates, due to careful
    identification.

It can lead to instability!
  •    Explanation will come.

15
MIT Rule Problem
  • Zp(s) is approximately known and modelled by
    Zm(s), km is known, kp is positive and unknown,
    but kc(t) is known and adjustable
  • Problem find a rule using e(t) to adjust kc(t)
    to cause e(t) to go to zero
  • Problem source kp depends on dynamic air
    pressure for aircraft.
  • Simple gradient descent algorithm (gain g) can be
    found

16
Example of performance
  • Unshaded region
  • is stable
  • Sine wave input
  • at frequency ?
  • Plant is (s1)-1

17
Performance second example
  • Unshaded region
  • is stable
  • Sine wave input
  • at frequency ?
  • Plant is e-s(s1)-1
  • while model is still
  • (s1)-1

18
Explaining Instability
  • First instability mechanism is interaction of
    excited plant dynamics with adaptive dynamics,
    made worse at high gain g
  • Problem is fixed by separating time scales
    (adapting slowly), with averaging theory
    justifying
  • Second separate instability mechanism comes from
    exciting at frequencies where Zp(s) and Zm(s) are
    significantly different.
  • No part of the adaptation mechanism was designed
    to deal with this.
  • Understanding of cause allowed technique to be
    safely used elsewhere, including radio telescope
    design in Australia.


19
Outline
  • Introduction
  • Some old problems of adaptive control
  • Generic and conceptual challenges
  • Impractical control objectives
  • Transient instability
  • Suddenly unstable closed loops
  • Changing experimental conditions
  • Topical Problems
  • Conclusions

20
Impractical objective
  • Plant is initially unknown or partially known, or
    is slowly varying.
  • There is an underlying performance index, e.g.
  •   The closed loop with known plant may have
    phase margin of 2.

The index may not be practically achievable
AND--separate point-- you may not know!
21
Impractical objective
The index may not be practically achievable
AND--separate point-- you may not know!
  • Good features of an adaptive control algorithm
    are
  • It will identify that an objective is infeasible
  • It will gracefully terminate
  • Windsurfer approach to adaptive control does this
  • It would be desirable if all adaptive control
    algorithms had this feature!!!

22
Transient instability
  • Theorem Consider the plant X, and the adaptive
    control law Y. Under conditions A,B,C, as t??,
    the parameter estimates do this nice thing, all
    signals are bounded, and performance approaches
    that of known plant case, or something similar.
  • This does not rule out
  • The plant input assuming a value of 106 before it
    settles down to its steady state value of 1
  • Nice bounds are hardly ever available
  • Editorial comment
  • It is misleading and it has been
    counterproductive to tell people (without
    qualification) that stability is proven
  • It would be desirable to prove practical bounded
    gain results and stop pretending the stability
    proofs are enough!!!

23
Suddenly unstable closed loops
  • Consider a plant-controller combination which
    suddenly goes unstable
  • May be due to a fault
  • May be due to connecting the wrong controller
  • The practical problem is to change the controller
    to eliminate the instability in a very short time
  • Virtually no adaptive control theorem considers
    this scenario
  • An additional complication is that the
    closed-loop signal may no longer be rich the
    instability dominates.
  • Old but still relevant data may save the day

24
Changing Experimental Conditions
  • In adaptive control, at each time instant
  • There is a model of the plant (which may be a
    good model)
  • There is a certain controller attached to the
    plant
  • If the plant model is a good one, a simulation of
    the model and controller will perform like the
    actual plant and controller
  • In the adaptive part of adaptive control
  • The controller may be changed to better reflect a
    control objective
  • The calculation of the new controller is based on
    the current model --applying with the current
    controller

25
Changing Experimental Conditions
Similar open-loop behaviours and
Open-loop
Closed-loop
26
Changing Experimental Conditions
  • Moral changing the controller may turn a good
    plant model into a bad one, or vice versa
  • If you change the controller significantly to
    suit the model, you might produce instability
    with the real plant,
  • This explains instability arising in iterative
    identification and controller redesign
  • It also explains it in Multiple Model Adaptive
    Control (later)
  • Safe adaptive control refers to
  • Ensuring you never, even temporarily, connect a
    destabilising controller
  • It may require you also to never, even
    temporarily, connect a controller which gives
    very poor performance
  • It would be desirable if all adaptive control
    algorithms had this safety feature!!!

27
Outline
  • Introduction
  • Some old problems of adaptive control
  • Generic and conceptual challenges
  • Topical Problems
  • Multiple Model Adaptive Control
  • Model Free Adaptive Control
  • Validating safety with closed-loop data
  • Conclusions

28
Multiple Model Adaptive Control
  • Imagine a bus on a city street. The equations of
    motion of the bus have parameters depending on
  • the load
  • The friction between tyres and road
  • In many plant models a (frequently small) number
    of physically-originating parameters are
    changeable/unknown.
  • Learning a parameter vector ? from measurements
    with an equation of the form
  • may be too hard, especially for nonlinear
    plants


29
Multiple Model Adaptive Control
  • An alternative approach (MMAC) is as follows
  • Suppose that the unknown parameter ? lies in a
    bounded simply connected region. Call the unknown
    plant .
  • Choose a set of values in this region, with
    associated plants P1,.......,PN.
  • Design (in advance) nice controllers for
    P1,.......,PN.
  • Call them C1,......,CN .
  • Run an algorithm which at any instant of time
    estimates (via the measurements) the particular
    Pi which is the best model to explain the
    measurements from . Call the associated
    parameter
  • Connect up

30
Multiple Model Adaptive Control
  •   Supervisor studies effect of using present
    controller and decides whether    or not to
    switch controller
  •   Desirable outcome after a finite number of
    switchings, the best controller    for the plant
    is obtained.

31
Why the name multiple model?
  • Underlying precept is that the plant
    coincides with or is near one of N nominal plants
    P1,.......,PN
  • Controller i, denoted Ci, is a good controller
    for Pi (and possibly plants near Pi)
  • All the preceding is valid in principle for
    nonlinear     plants!

32
Deciding the Best Model for P
Multi- estimator
LINEAR PLANTS NOW!
  • u y1
  • Multi-
  • y estimator yN
  • r Controller k
    Plant P
  • -
  • Multi-estimator is a device which produces N
    outputs
  • if (and only if with complicated
    signals)
  • (The controller is irrelevant in this
    condition)

_ Plant P
Controller k
33
Supervision using Multi-estimator
  • Multi- y1
  • estimator
  • y yN
  • r Controller J
    Plant P
  • -
  • Idea of algorithm study
  • for some small a gt 0, and k1,,N. If the
    smallest occurs for
  • k I, say that P is best modelled by
  • Switch in

This may lead to switching in a destabilising
controller!
34
Example
  • Plant is 3rd order, stable, with non-minimum
    phase zero in 1,10 and DC gain in .2,2.
  • Control objective extend bandwidth beyond open
    loop plant, with closed loop transfer function
    close to 1 in magnitude.
  • Non-minimum phase zero is a limiter.
  • 441 plant models chosen, with DC gain and
    non-minimum phase zero each in 20 logarithmically
    space intervals.
  • Reference signal is wideband noise
  • Measurement noise and process (input) noise are
    present

35
Example of Temporary Instability
36
Example of Temporary Instability
37
Multiple Model Adaptive Control Difficulties
  • Should there be 7, 70 or 7000 models? How should
    one actually choose the models?
  • How can one avoid the (temporary) instabilities?

These questions are actually linked.
  •   They can be systematically resolved using
    notions of a ?-gap     metric (tested on the 441
    model case) or performance-based     tools
    developed by Fekri and Athans 

38
Avoiding Instability Safe Switching
  • P may be best modelled by PI when CJ is used, but
    best modelled by PK when CI is used.
  • PI may in fact be a terrible model of P when CI
    is used.
  • Nontrivial fact using crude estimation
    techniques one can obtain set of controllers CK
    which will retain stability.
  • can even retain similarity of performance.
  • Tool for this is Vinnicombe metric
  •     WARNING TO USER If you switch controllers
    very frequently, you       can lose  stability.

39
Model Free Adaptive Control
  • Rule 1 Never try to estimate the plant, which
    may be nonlinear. Big claim 1You dont even need
    a model.
  • Use a finite set of controllers, where one at
    least is guaranteed satisfactory (stability and
    performance) for any plant that could be
    encountered
  • At any instant of time evaluate performance with
    current controller and other possible
    controllers
  • Determine external input that would have to be
    used with each candidate controller to get SAME
    plant input and output as actual
  • Compute the performances for each case and
    compare.
  • Switch when current performance is not best.
  • First catch (repairable) controllers must be
    invertible (minimum phase and bi-proper)
  • Other catches follow.

40
Model Free Adaptive Control
  • Rule 1 Never try to estimate the plant, which
    may be nonlinear. Big claim 1 You dont even
    need a model.
  • Use a finite set of controllers, where one at
    least is guaranteed satisfactory (stability and
    performance) for any plant that could be
    encountered
  • You cant get a finite set of controllers where
    one at least is fine without knowing something
    about the plant. Model Free is a misnomer in
    this sense.
  • At any instant of time evaluate performance with
    current controller and performances that would
    result with each other possible controller,
  • if the external input were such that the same
    plant input and output resulted.
  • Switch when current performance is not best.
  • It is doubtful that one would wish to compare
    performances for different external inputs, and
    identical plant inputs and outputs.

41
Model Free Adaptive Control
  • At any instant of time evaluate performance with
    current controller and performances that would
    result with each other possible controller,
  • if the external input were such that the same
    plant input and output resulted.
  • Switch when current performance is not best.
  • It is doubtful that one would wish to compare
    performances for different external inputs, and
    identical plant inputs and outputs.
  • You CANNOT check that other controllers are or
    are not stabilising. You can only actually
    evaluate performance in advance IF the possible
    controller is stabilizing.
  • The method can even result in REPEATED connection
    of the same destabilizing controller, in
    aggregate for a long time.
  • It illustrates again the need for safe adaptive
    control--i.e. ability to confirm before switching
    that a new controller will not destabilise.

42
Validating safety what would be nice
  • On the basis of noisy measurements taken on the
    unknown plant with a known controller, be able to
    say with confidence whether a new controller will
    be stabilizing or not.
  • New results are becoming available. They rely on
    obtaining noisy frequency response data and using
    a Nyquist-like diagram.
  • Multivariable version available--consider here
    single-input, single-output.

43
Validating stability with new controller
May be OK to have C1 and C2 in forward or
feedback paths only.
  • C1 is stabilizing
  • C2 is candidate to replace C1
  • Experiment gives transfer function T from r to
    z (no stability     problem, will need richness,
    can have noise)
  • C2 is stabilizing (in place of C1) if Nyquist
    plot of T does  not     encircle origin.

44
Validating Safety with Closed-loop data (cont.)
  • The validation test uses limited amount of noisy
    closed-loop data to guarantee P, C2 will be
    stable
  • It assumes no a priori information about the
    plant
  • except linear and time-invariant
  • It can potentially address transient instability
    problem
  • The experiments can tolerate significant error
  • It uses the phase response only up to a finite
    frequency
  • A small controller change implies a smaller
    frequency band over which T must be estimated

45
Outline
  • Introduction
  • Some old problems of adaptive control
  • Generic and conceptual challenges
  • Topical Problems
  • Conclusions

46
Conclusions
  • The common but not the only thread has been
    unplanned instability--but the causes are all
    different.

Bursting Rich excitation is needed else turn off adaptation
Iterative Identification and Controller Redesign Dont make big controller changes. Check safety.
MIT rule Keep three time scales separate Dont rely on adaptation to take care of sloppy modelling
Rohrs counterexample Bursting and MIT conclusions
Impractical control objectives Allow for impractical objective with graceful failure mechanism
47
Conclusions
  • The common but not the only thread has been
    unplanned instability--but the causes are all
    different.

Transient instability Stability as t ! 1 is not enough
Dealing with instability A big problem!
Changing experimental conditions/controller Use safe adaptive control principles/small changes
Multiple model adaptive control Very promising for nonlinear. Need systematic modelling/safe adaption
Model free adaptive control Fails a number of tests, and delivers less than might seem promised.
Checking safety Noisy Nyquist diagram may help
48
Macro-view of presentation
  • There are big mistakes revealed in Adaptive
    Control history, especially by theoreticians
  • There are pervasive adaptive control problems
    still not properly addressed, of great practical
    significance.

49
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